We demonstrate fast two-qubit gates using a parity-violated superconducting qubit consisting of a capacitively shunted asymmetric Josephson-junction loop under a finite magnetic flux bias. The second-order nonlinearity manifesting in the qubit enables the interaction with a neighboring single-junction transmon qubit via first-order interqubit sideband transitions with Rabi frequencies up to 30 MHz. Simultaneously, the unwanted static longitudinal ($ZZ$) interaction is eliminated with ac Stark shifts induced by a continuous microwave drive near resonant to the sideband transitions. The average fidelities of the two-qubit gates are evaluated with randomized benchmarking as 0.971, 0.958, and 0.962 for CZ, iswap, and swap gates, respectively.
